Questions tagged [deformation-theory]
for questions about deformation theory, including deformations of manifolds, schemes, Galois representations, and von Neumann algebras.
643
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A general definition of an equisingular family of singular varieties?
This question is about the existence of a definition. I'm far from being an expert in the field in question I apologize in advance for any inaccuracies or stupid and wrong assumptions.
Let $X$ be a ...
5
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1
answer
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Hodge numbers of a Calabi-Yau 3-fold via deformation theory
In their paper "Calabi-Yau 3-folds and Moduli of Abelian Surfaces I", Gross and Popescu calculate the hodge numbers of smooth CY 3-folds obtained in the following way (see Remark 4.11 of their paper): ...
4
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Does the local Bertini theorem in mixed characteristic imply the global Bertini theorem
Let $R$ be a discrete valuation ring with fraction field $K$ and residue field $k$. Assume that the characteristic of $K$ is $0$ and of $k$ is $p>0$. Let $\pi:X \to \mbox{Spec}(R)$ be a flat, ...
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0
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On regularity of flat families over a DVR
Let $k$ be an algebraically closed field of characteristic zero and $R$ a discrete valuation ring over $k$. Let $\pi:X \to \mathrm{Spec}(R)$ be a flat, projective morphism such that the generic fiber ...
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2
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Deformations of Calabi-Yau manifolds
Let $X$ be a compact complex smooth manifold with holomorphically trivial canonical class.
It is true that any (sufficiently small?) deformation of the complex structure of $X$ also has ...
4
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Building conilpotent coalgebras from co-square-zero-extensions
Let $\mathrm{K}$ be a field of char. 0.
Given a chain complex $\mathrm{X} $ over $\mathrm{K}$ denote $\mathrm{E}(\mathrm{X})$ the co-square-zero-extension on $\mathrm{X}, $ i.e. the
cocommutative ...
3
votes
1
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Lie algebras : Deformations and Rigidity
I am studying deformation (as it is introduced in https://arxiv.org/pdf/math/0611793.pdf or http://web.cs.elte.hu/~fialowsk/pubs-af/condefnew2.pdf) and rigidity of some infinite dimensional Lie ...
3
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2
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Deformation of "Hecke modification"
Let $X$ be a smooth curve over $\mathbb{C}$. I wish to compute the deformation of the following data $(E,F,x)$. $E$ and $F$ are locally free sheaves over $X$ and $x$ is a point on $X$. They satisfy:
...
9
votes
1
answer
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deformation theory in positive characteristic
The idea "Formal deformation theory in characteristic zero is controlled by a differential graded Lie algebra (dgla)" goes back to Goldman-Millson, Deligne, Drinfeld among others; see Lurie's ICM talk....
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Compute action of the gauge group in deformation theory of an algebra
I am working on Balazs Szendroi's introduction to deformation theory, but I got stuck on the exercise on the bottom of page 6.
Consider a vector space $A$ with a multiplication $m$ that makes it into ...
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Exercise 1.1.(c) in Hartshorne's Deformation Theory
Exercise 1.1.(c) in Hartshorne's Deformation Theory:
Over an algebraically closed field $k$, we define a curve in $\mathbb P^2_k$ to be the closed subscheme, defined by a homogeneous polynomial $f(...
1
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1
answer
334
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Holomorphic family of Riemann surfaces
Let $2g+m\ge 3$. A holomorphic family of (non-singular, compact) Riemann surfaces of type $(g,m)$ is a triple $(X,Y,\pi,s_1,\ldots,s_m)$, where $X,Y$ are complex manifolds of (complex) dimension $n+1,...
3
votes
1
answer
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Degeneration of coadjoint orbits
Let $X$ be a projective manifold and we have a degeneration of fibers such that they are biholomorphic to coadjoint orbits, i.e, $X\to \Delta$ , and fibers $X_t$ are biholomorphic to coadjoint orbits ...
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Tangent space in Algebraic geometry and Differential geometry
We know in differential geometry, given a $C^k$ manifold for $k>1$, the tangent space at a point in this manifold is parametrized by curves passing through this point modulo certain equivalence ...
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What is a deformation of a category?
I have several naive and possibly stupid questions about deformations of categories. I hope that someone can at least point me to some appropriate references.
What is a deformation of a (linear, dg, ...
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Introduction to deformation theory (of algebras)?
So I know that the idea of deformation theory underlies the concept of quantum groups; I haven't found any single introduction to quantum groups that makes me fully satisfied that I have some kind of ...
9
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Degeneration of curves inside a family of surfaces
We are interested in understanding how a higher genus curve in a smooth surface of general type can degenerate into the non-normal locus of the limit of a degeneration of surfaces. More precisely:
...
2
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1
answer
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Identification of cohomology sheaf in the definition of the Kodaira-Spencer morphism for abelian schemes
Let $p:A \to S$ be a projective abelian scheme, where $S$ is some smooth scheme over a base field $k$. Then we have the Kodaira-Spencer morphism
$$
\kappa : T_{S/k} \to R^1p_*T_{A/S}
$$
where $T_{S/k}$...
3
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1
answer
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Deformation Theoretic Interpretation of $H^1(C,T_C(-2p))$
Suppose $C$ is a (non-singular) compact Riemann surface of genus $g$ and with $n$ (distinct) marked points $p_1,\ldots,p_n$. If we assume the stability condition ($2-2g-n<0$), then it is proved in "...
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Finite dimensional approximation of Donaldson theory
In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...
3
votes
1
answer
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A differential graded Lie algebra with the Hochschild differential
Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following ...
9
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"A theory of generalized Donaldson-Thomas invariants" by Joyce & Song
Is anyone else working through this paper: A theory of generalized Donaldson-Thomas invariants, by Dominic Joyce, Yinan Song? I am trying to verifying example 6.2 (m=2 for simplicity) using only the ...
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Is Hironaka's example the only known deformation of Kähler manifolds with non-Kähler central fibre?
A well-known example in the deformation theory of compact complex manifolds is the one given by Hironaka in his 1962 paper An Example of a Non-Kählerian Complex-Analytic Deformation of Kählerian ...
8
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Deformation-Obstruction Theory of YM Instantons
In Donaldson-Kronhiemer Section 4.2.5. (local models of the moduli space of YM instantons) they first get local models of the moduli space $M$ inside the space of all connections modulo gauge $\...
3
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Vanishing of space of first order infinitesimal deformations for irreducible algebraic stack
This question has a few bits, and apologies if some questions are phrased poorly since I am not knowledgeable on the language of stacks or deformation theory.
Suppose $\mathscr{X}$ is an algebraic ...
0
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1
answer
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Can two singular points collapse to a new singular point?
It is about deformation theory on algebraic surfaces.
If there are two singular points on an algebraic surfaces, is it possible that two singular points collapse to a new point as the surface deforms?
...
0
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1
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347
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Is the "addition" of flat morphisms flat?
Let $f:X \to Y$ be a flat, projective morphism between projective varieties. Let $F, G$ be coherent sheaves on $X$, flat over $Y$. Let $\phi_1, \phi_2$ be two morphisms from $F$ to $G$ such that:
1) ...
9
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Clarification on relationship between Grothendieck-Messing and Honda systems
It's well-known, due to work of Fontaine (see e.g. this), that if $k$ is a perfect field of characteristic $p$, then one can classify $p$-divisible groups over $W(k)$ by Honda systems. Namely, these ...
4
votes
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3-Gerstenhaber algebra structure on the cohomology of deformation complexes?
In a seminal paper "On the Deformation of Rings and Algebras", M. Gerstenhaber showed that the deformation complex of any associative algebra (known as the Hochschild complex) is naturally endowed ...
3
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Equivalence of deformations of non-associative algebras
Let $(\mathcal A,\mu)$ be an associative algebra. According to usual deformation theory, deformations of $(\mathcal A,\mu)$ as an associative algebra are controlled by a differential graded algebra (...
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Kodaira-Spencer maps and deformation theory
This post concerns the following question: Can we black-box the analysis of PDE's which arises in the construction of Kuranishi families for complex analytic structures?
The deformation theory of ...
3
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answers
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Obstruction to lifting coherent sheaves on discrete valuation ring
Let $R$ be a discrete valuation ring with algebraically closed residue field $k$. Let $K:=\mathrm{Frac}(R)$ the fraction field of $R$. Suppose $K$ is of characteristic zero. Denote by $\overline{K}$ ...
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vote
1
answer
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Isomorphism of sheaves in families of projective varieties
Let $\pi:\mathcal{X} \to S$ be a flat, family of projective varieties (here $\mathcal{X}$ and $S$ are noetherian). Let $E$ and $F$ be two locally free sheaves on $\mathcal{X}$ such that for all $s \in ...
0
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0
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Vector bundles on families of rational curves
Let $\pi:\mathcal{X} \to S$ be a flat, projective family of rational curves ($S$ is noetherian) over an algebraically closed field $k$. Assume $S$ is irreducible. Let $E$ be a locally-free sheaf on $\...
6
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A question on deformation theory of triples of matrices
Let $(x,y,z)$ be a triple of $n \times n$ traceless complex matrices which are simultaneously diagonalizable. We call such a triple regular if $C_x \cap C_y \cap C_z$ is a Cartan subalgebra of $\...
4
votes
1
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Projection formula for field extension
Let $X$ be a projective variety over a field $K$ of characteristic zero. Denote by $p:X_{\overline{K}} \to X$ the natural morphism, where $\overline{K}$ is the algebraic closure of $K$ and $X_{\...
3
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Examples of varieties with every stable sheaf simple
Are there examples of projective varieties over a non-algebraically closed field such that every geometrically stable sheaf on the variety is simple? I see, for example in Huybrechts-Lehn and in some ...
2
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Base change, descent theory and coherent sheaves
Let $k$ be a field of characteristic zero and $X$ a smooth, projective $k$-variety. Let $E_{\overline{k}}$ be a coherent sheaf on $X_{\overline{k}}$ ($\overline{k}$ denotes the algebraic closure of $k$...
6
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Are two Lie algebra deformations with cohomologous tangents isomorphic?
Let $(\mathfrak{g},[-,-])$ be a finite-dimensional Lie algebra. I'm interested in real Lie algebras, but feel free to complexify if this helps.
Say I'm interested in classifying isomorphism ...
3
votes
1
answer
659
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Deformation of a Hopf algebra
A deformation of a Hopf algebra is defined as follows.
On page 171 of the book a guide to quantum groups, Remark 2, it is said that Any deformation of a Hopf algebra $A$ as a bialgebra is ...
0
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1
answer
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Can the specialization map be flat
Let $X$ be a projective variety over an algebraically closed field of characteristic zero. Let $\eta$ be a generic point of $X$ and $x$ be a closed point. By http://stacks.math.columbia.edu/tag/054F ...
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Functoriality of the formality quasi-isomorphism of E-polydifferential operators
Given a smooth manifold $M$ and a Lie algebroid $E\rightarrow M$ we can consider the $E$-polydifferential operators $D_E$ and the $E$-polyvectorfields
$T_E$ as
$$D_E:=\bigoplus_{k=-1}^\infty\mathcal{...
12
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What does deformation theory have to do with Serre duality?
The cotangent bundle $\Omega_X$ of a smooth scheme $X$ shows up in two places in my understanding of algebraic geometry. The first is deformation theory, where maps out of $\Omega_X$ control the ...
3
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0
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Families of unbounded operators
Let $H$ be a Hilbert space, $X$ a topological space, and $\{A_t\}_{t\in X}$ a continuous family of bounded, invertible operators on $H$. Continuous here in the sense that the corresponding map $X\...
5
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DGLA related to the deformation of hopf Algebras
Recently I was considering Hopf algebras and Drinfeld's twists. I stumbled upon
a certain DGLA one can associate to a Hopf algebra (unital bialgebras actually) by copying the formulas obtained by ...
13
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3
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DGLA or $L_{\infty}$-algebra controlling the deformation of Einstein metrics and instantons
As proposed by Quillen, Drinfeld, and Deligne and other important mathematicians, there is supposed to be a philosophy that, at least over a field of characteristic zero, assigns to every "deformation ...
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Is complete intersection a open or closed property in Hilbert schemes
Fix an integer $N$, $X$ a (smooth) complete intersection subvariety in $\mathbb{P}^N$. Denote by $P$ the Hilbert polynomial of $X$ (as a subvariety in $\mathbb{P}^N$). Consider the Hilbert scheme $\...
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0
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Generic properties of families of algebras over an infinite dimensional base space
Let $\mathbb{k}$ be an algebraically closed field and let $A$ be a $\mathbb{k}$-algebra which is a free module of rank $r$ over some central subalgebra $Z_0$. If $Z_0$ is affine and $r$ is a finite ...
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Could we extend the star product on a Poisson manifold from its ring of smooth functions to its de Rham complex?
Let $M$ be a smooth manifold with a Poisson bracket $\{-,-\}$. Kontsevich proved that there exists a deformation quantization of $M$, i.e. let $C^{\infty}(M)[[\hbar]]=C^{\infty}(M)\otimes_{\mathbb{R}}\...
3
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1
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Reference for a folklore result about $T^1(B/k;M)$
If $B$ is a $k$-algebra, let $T^1(B/k;M)$ denote the first cotangent functor. It classifies first order deformations of the scheme $\mathrm{Spec} B$.
Now, if $X \subset \mathbb P^n$ is a smooth ...